152 155] GKEEN'S FUNCTION. 295 



and for all points lying on the plane 2, 



r r n r 2 



Consequently the function 



vanishes for points on either plane, and being harmonic except 

 at P, is Green's function. 



154. Two parallel Planes. If P lie in the space between 

 two parallel planes the successive images will all lie in a straight 

 line, and will be infinite in number. Using the same notation as 

 in the last example, we have the same equations, and the same 

 form of Green's function, except that we shall have an infinite 

 series. 



155. Sphere. Let A be the given pole, at a distance a from 

 the center of the sphere of radius R. Take a point B lying on the 

 same radius as A, at a distance from the center 6 such that ab=R z . 

 Then A and B are said to be inverse points with respect to the 

 sphere. If M be any point on the surface of the sphere, the 

 triangles OMB and MAO are similar, for they have a common 

 angle at 0, and the sides including it are proportional, for by 

 hypothesis, 



FIG. 63. 



Accordingly, for points on the surface 



r' r . I Rl 

 < 2 > E = a and 7 == a/ 



